So far, filtrations and their persistent homology are defined as abstract entities. When the filtration consists of unions of balls, nerve lemma (Theorem 2.3 of [1]}) guarantees it can be algorithmically computed. This is not the case for the distance-to-measure filtration as well as the proposed filtration. In the present work, the proposed filtration is approximated by a cubical filtration. A cubical filtration is a filtration consisting of unions of points, line segments, squares, cubes and their higher-dimensional analogues. Since these topological spaces are made up of parts that can be exhaustively enumerated, their persistent homology can be efficiently computed. An efficient algorithm to compute the persistent homology of a cubical filtration is proposed in [2]} and it is used in the present work.
